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  {
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    "# Attention Pooling: Nadaraya-Watson Kernel Regression\n",
    "\n"
   ]
  },
  {
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   "execution_count": 1,
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     "pytorch"
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   },
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   "source": [
    "import torch\n",
    "from torch import nn\n",
    "from torch.nn import functional as F\n",
    "from d2l import torch as d2l"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Generating the Dataset"
   ]
  },
  {
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   "execution_count": 4,
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     "id": "",
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    "tab": [
     "pytorch"
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    "class SinData(d2l.DataModule):\n",
    "    def __init__(self, n, batch_size):\n",
    "        self.save_hyperparameters()\n",
    "        f = lambda x: 2 * torch.sin(x) + x**0.8\n",
    "        self.x_train, _ = torch.sort(torch.rand(n) * 5)\n",
    "        self.y_train = f(self.x_train) + torch.randn(n)\n",
    "        self.x_val = torch.arange(0, 5, 5.0/n)\n",
    "        self.y_val = f(self.x_val)\n",
    "\n",
    "    def get_dataloader(self, train):\n",
    "        arrays = (self.x_train, self.y_train) if train else (self.x_val, self.y_val)\n",
    "        return self.get_tensorloader(arrays, train)\n",
    "\n",
    "n = 50\n",
    "data = SinData(n, batch_size=10)\n",
    "\n",
    "def plot_kernel_reg(y_hat):\n",
    "    d2l.plot(data.x_val, [data.y_val, y_hat.detach().numpy()], 'x', 'y', legend=['Truth', 'Pred'],\n",
    "             xlim=[0, 5], ylim=[-1, 5])\n",
    "    d2l.plt.plot(data.x_train, data.y_train, 'o', alpha=0.5);\n",
    "\n",
    "y_hat = data.y_train.mean().repeat(n)\n",
    "plot_kernel_reg(y_hat)"
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  {
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   },
   "source": [
    "Nonparametric Attention Pooling"
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  },
  {
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   "execution_count": 5,
   "metadata": {
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     "id": "",
     "n": "40"
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    "origin_pos": 12,
    "tab": [
     "pytorch"
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      "text/plain": [
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     },
     "metadata": {
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     "output_type": "display_data"
    }
   ],
   "source": [
    "def diff(queries, keys):\n",
    "    return queries.reshape((-1, 1)) - keys.reshape((1, -1))\n",
    "\n",
    "def attention_pool(query_key_diffs, values):\n",
    "    attention_weights = F.softmax(- query_key_diffs**2 / 2, dim=1)\n",
    "    return torch.matmul(attention_weights, values), attention_weights\n",
    "\n",
    "y_hat, attention_weights = attention_pool(\n",
    "    diff(data.x_val, data.x_train), data.y_train)\n",
    "plot_kernel_reg(y_hat)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Attention weights"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "attributes": {
     "classes": [],
     "id": "",
     "n": "41"
    },
    "origin_pos": 14,
    "tab": [
     "pytorch"
    ]
   },
   "outputs": [
    {
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\" y=\"-9.483219\"/>\n",
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    "d2l.show_heatmaps([[attention_weights]],\n",
    "                  xlabel='Sorted training inputs',\n",
    "                  ylabel='Sorted testing inputs')"
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    "Given two tensors of shape ($n$, $a$, $b$) and ($n$, $b$, $c$), the shape of their batch matrix multiplication output is ($n$, $a$, $c$).\n",
    "Use minibatch matrix multiplication to compute weighted averages of values in a minibatch.\n",
    "Parametric attention pooling"
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  {
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   "execution_count": 7,
   "metadata": {
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    "class NWKernelRegression(d2l.Module):\n",
    "    def __init__(self, keys, values, lr):\n",
    "        super().__init__()\n",
    "        self.save_hyperparameters()\n",
    "        self.w = torch.ones(1, requires_grad=True)\n",
    "    def forward(self, queries):\n",
    "        y_hat, self.attention_weights = attention_pool(\n",
    "            diff(queries, self.keys) * self.w, self.values)\n",
    "        return y_hat\n",
    "\n",
    "    def loss(self, y_hat, y):\n",
    "        l = (y_hat.reshape(-1) - y.reshape(-1)) ** 2 / 2\n",
    "        return l.mean()\n",
    "\n",
    "    def configure_optimizers(self):\n",
    "        return d2l.SGD([self.w], self.lr)"
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    "Transform the training dataset\n",
    "to keys and values"
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   "source": [
    "model = NWKernelRegression(data.x_train, data.y_train, lr=1)\n",
    "model.board.display = False\n",
    "trainer = d2l.Trainer(max_epochs=5)\n",
    "trainer.fit(model, data)\n",
    "\n",
    "plot_kernel_reg(model.forward(data.x_val))"
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  },
  {
   "cell_type": "markdown",
   "metadata": {
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   },
   "source": [
    "The region with large attention weights becomes sharper"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "origin_pos": 21,
    "tab": [
     "pytorch"
    ]
   },
   "outputs": [
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    "d2l.show_heatmaps([[model.attention_weights]],\n",
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